cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A265740 a(1)=1; a(n+1) is the smallest positive integer not yet used such that all the digits of a(n) and a(n+1) are present in the decimal expansion (including any leading and trailing zeros) of a(n)/a(n+1).

Original entry on oeis.org

1, 6, 13, 10, 14, 17, 7, 8, 19, 23, 21, 29, 34, 31, 3, 38, 28, 46, 47, 35, 39, 49, 43, 51, 42, 41, 48, 53, 26, 12, 57, 58, 59, 2, 61, 24, 68, 11, 52, 63, 22, 69, 62, 71, 56, 65, 76, 81, 44, 67, 64, 83, 85, 78, 77, 79, 72, 70, 80, 87, 84, 86, 89, 9, 91, 92, 73
Offset: 1

Views

Author

Eric Angelini, submitted by Lars Blomberg, Dec 15 2015

Keywords

Comments

Conjecture: a(n) is a permutation of the natural numbers.
The following table shows:
C = number of terms calculated
F = first term that is missing
C F F/C
1000 5 0.005
2000 50 0.025
5000 1650 0.330
10000 1650 0.165
20000 2475 0.124
50000 24750 0.495
100000 100000 1.000
200000 199800 0.999
500000 499500 0.999
which seems to support the conjecture.

Examples

			   1/6  = 0.1666... (1 and 6 are visible on the right-hand side)
   6/13 = 0.461538461538... (6, 1 and 3 are visible)
  13/10 = 1.30 (trailing zeros are included)
  10/14 = 0.7142857142... (1, 0 and 4)
  14/17 = 0.8235294117... (1, 4 and 7)
  17/7  = 2.4285714285... (1 and 7)
   7/8  = 0.875 (7 and 8)
   ...

Links

Crossrefs

See A265756 for another version.
See also A257664.

Programs

  • Mathematica
    f[n_] := Block[{a = {1}, k}, Do[k = If[MissingQ@ #, Max@ a, #] &@ SelectFirst[Range@ Max@ a, ! MemberQ[a, #] &]; While[Or[! AllTrue[Join[IntegerDigits@ a[[i - 1]], IntegerDigits@ k], MemberQ[Union@ Flatten@ Prepend[First@ #, If[Last@ # <= 0, 0, Nothing]] &@ If[Depth@ First@ # < 3, Insert[#, 0, {1, 1}], #] &@ RealDigits[a[[i - 1]]/k], #] &], MemberQ[a, k]], k++]; AppendTo[a, k], {i, 2, n}]; a]; f@ 67 (* Version 10.2 *)
f[n_] := Block[{a = {1}, k}, Do[k = 1; While[Or[If[# == 1, False, True] &[Times @@ Boole[MemberQ[Union@ Flatten@ Prepend[First@ #, If[Last@ # <= 0, 0]] &@ If[Depth@ First@ # < 3, Insert[#, 0, {1, 1}], #] &@ RealDigits[a[[i - 1]]/k], #] & /@ Join[IntegerDigits@ a[[i - 1]], IntegerDigits@ k]]], MemberQ[a, k]], k++]; AppendTo[a, k], {i, 2, n}]; a]; f@ 67 (* Michael De Vlieger, Dec 16 2015, Version 6 *)

Extensions

Corrected values for n>=58 by Lars Blomberg, Dec 16 2015

A265756 a(1)=1; a(n+1) is the smallest positive integer not yet used where all the digits of a(n) and a(n+1) are present in the decimal expansion (excluding any leading or trailing zeros) of a(n)/a(n+1).

Original entry on oeis.org

1, 6, 13, 17, 7, 8, 14, 19, 23, 21, 29, 34, 31, 3, 38, 28, 46, 47, 35, 39, 49, 43, 51, 42, 41, 48, 53, 26, 12, 57, 58, 59, 2, 61, 24, 68, 11, 52, 63, 22, 69, 62, 71, 56, 65, 76, 81, 44, 67, 64, 83, 85, 78, 77, 79, 72, 70, 87, 80, 89, 9, 86, 92, 73, 27, 84, 93
Offset: 1

Views

Author

Hans Havermann, Dec 15 2015

Keywords

Comments

Conjecture: a(n) is a permutation of the natural numbers.

Examples

			1/6 = .1666... (1 and 6)
6/13 = .461538461538... (1, 3 and 6)
13/17 = .76470588235294117647058823529411... (1,3 and 7)
Note that term 4 is not 10 because 13/10 = 1.3 (no zero)

Links

Crossrefs

Cf. A257664, A265740.

Programs

  • Mathematica
    t=1; s={1}; Do[c=1; d=IntegerDigits[t]; While[f=Flatten[RealDigits[t/c][[1]]]; u=Union[IntegerDigits[c], d]; Union[Table[MemberQ[f, u[[i]]], {i, Length[u]}]]!={True}||MemberQ[s, c], c++]; t=c; AppendTo[s, t], {400}]; s

A266281 a(1)=1; a(n) is the first integer > a(n-1) with which, in the a(n-1)/a(n) decimal expansion, n is present.

Original entry on oeis.org

1, 4, 11, 13, 14, 17, 19, 22, 23, 29, 34, 47, 58, 61, 65, 87, 89, 93, 94, 97, 102, 103, 105, 109, 113, 115, 116, 118, 121, 130, 131, 136, 139, 141, 149, 152, 157, 159, 161, 166, 167, 169, 174, 177, 179, 181, 184, 188, 191, 193, 194, 197, 199, 203, 218, 223, 224
Offset: 1

Views

Author

Francesco Di Matteo, Dec 26 2015

Keywords

Comments

Two other sequences are possible without the a(n)>a(n-1) limitation, one with a(n) terms already used in the sequences (where the n growth does not allow data looping), another only with a(n) terms not yet used.

Examples

			a(8) = 22 because a(7) = 19 and "8" does not appear in the digital expansion of 19/20 = 0.95 nor of 19/21 = 0.904761904761..., but it does appear in 19/22 = 0.86363...;
a(9) = 23 because 22/23 = 0.9565217391304..., where "9" does appear;
a(10) = 29 because "10" does not appear in the digital expansion of 23/k for k=24..28, but it does appear in 23/29 = 0.7931034...

Links

Crossrefs

Cf. A257664, A265740, A265756.

Programs

  • Mathematica
    f[n_] := Block[{a = {1}, k}, Do[k = a[[m - 1]] + 1; While[SequenceCount[Flatten@ First@ RealDigits[a[[m - 1]]/k], IntegerDigits@ m] < 1, k++]; AppendTo[a, k], {m, 2, n}]; a]; f@ 57 (* Version 10.1, or *)
f[n_] := Block[{a = {1}, k}, Do[k = a[[m - 1]] + 1; While[StringCount[
ToString[FromDigits@ Flatten@ First@ RealDigits[a[[m - 1]]/k]], ToString@ m] < 1, k++]; AppendTo[a, k], {m, 2, n}]; a]; f@ 57 (* Michael De Vlieger, Dec 30 2015, Version 5.1 *)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.